an:01331418
Zbl 0945.45004
Lin, Yanping; Liu, James H.
Stability of nonlocal diffusion equations
EN
Dyn. Contin. Discrete Impulsive Syst. 5, No. 1-4, 53-66 (1999).
00053804
1999
j
45K05 45M10 65R20
stability; nonlocal diffusion equations; partial integro-differential equation; backward Euler difference methods
The partial integro-differential equation
\[
u_t + Au + \int_0^t K(t-s) Bu(s)ds + f(u) = 0, \quad 0 < t \leq T,
\]
is studied on a smooth domain \(\Omega\) with zero boundary conditions. The operator \(A\) is a strongly elliptic operator, so that the equation is parabolic in the absence of the integral term. The operator \(B\) is a second order differential operator with respect to \(x\), and \(K\) is a scalar nonnegative kernel. A number of theorems is proved on exponential decay and time-discretization by backward Euler difference methods. In addition, the well-posedness and time-discretization of this problem are studied under a nonstandard non-local time weighted initial condition
\[
u(x,0) = \sum_{k=1}^M \beta_k(x)u(x,T_k) + \psi(x), \quad x \in \Omega,
\]
where \(0 < T_1 < T_2 \cdots < T_M = T\).
O.Staffans (??bo)